You are here

Mathematical Analysis: A Concise Introduction

Publisher:

John Wiley

Number of Pages:

562

Price:

95.00

ISBN:

9780470107966

This is the Greatest Hits version of mathematical analysis. It covers all the high points of real and abstract analysis, but doesn't go into depth on anything. The treatment is rigorous and error-free. Roughly half the book deals with real analysis theorems, and half with abstract analysis and function spaces (including multivariable calculus). There is also a short section on physical applications, dealing mostly with differential equations.

The book skimps on worked examples. Most of the illustrative examples are in the exercises, often with hints. Many traditional analysis topics appear only in the exercises, for example, Stirling's formula for n!, Riemann-Stieltjes integrals, Lagrange multipliers, and the Stone-Weierstrass Theorem,

It may seem peculiar to subtitle a 562-page book "concise", but that is an accurate summary of the approach. The book manages to cover an enormous amount of material in those pages.

As I was reading I was continually puzzling over where the book would fit in the curriculum. The author used the draft in a two-quarter course at his university, but it must have been a struggle to get through all the material in that time. In some ways the book looks like a text for an introduction to proofs course, but it also includes a number of topics that would normally be in graduate courses, and everything in between. Is this "The Only Analysis Book You'll Ever Need"? Probably not — there's no complex analysis, for example, and Fourier analysis is only touched on briefly.

My big gripe with this book is that it is uninspiring. It never shows you why people get excited about analysis. The history of real analysis is largely a history of pathological cases and counterexamples, and how analysts prevailed over these difficulties to create a beautiful theory. This book shows you the theory but not the beauty.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.